2013 IEEE Wireless Communications and Networking Conference (WCNC): PHY

Secure Beamforming for MIMO Two-Way Transmission with an Untrusted Relay Jianhua Mo∗† , Meixia Tao∗ , Yuan Liu∗ , Bin Xia∗ and Xiaoli Ma†

∗ Department

† School

of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, P. R. China of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA Email: {mjh, mxtao, yuanliu, bxia}@sjtu.edu.cn, [email protected]

Abstract—From security perspective, a friendly relay may help to keep the confidential messages from being eavesdropped, while an untrusted relay may intentionally eavesdrop the messages when relaying. This paper studies the secure beamforming for multiple-input multiple-output (MIMO) two-way communications, where two source nodes exchange information with the help of an untrusted relay node. The relay adopts amplify-andforward (AF) strategy and acts as both an essential helper and a potential eavesdropper. Our goal is to maximize the secrecy sum rate of the bidirectional links by jointly optimizing the source and relay beamformers. For the two-phase two-way relay scheme, we first derive the optimal structure of the relay beamformer and then propose an iterative algorithm to jointly optimize the source and relay beamformers. Then, a comprehensive study on the asymptotical performance is conducted by letting the source and relay powers approach zero or infinity. In particular, we show that when all powers approach infinity, the two-way relay scheme achieves the maximum secrecy rate if the transceiver beamformers are designed such that the received signals at the relay can be aligned to be parallel.

I. I NTRODUCTION Security in wireless networks is traditionally considered at upper layers using cryptographic approaches. As the rapid increase of computational capability of smart devices, it is hard to guarantee the security requirements using these upper layer approaches only. To address this issue, the informationtheoretic security at the physical layer, which originates from Shannon’s notion of perfect secrecy, has recently attracted considerable attention [1]. Cooperative relaying can offer many benefits, such as power reduction and coverage extension. Therefore, it is attractive to utilize these benefits for physical layer security. According to the relay being trusted or untrusted, the related work can be divided into two categories. In the former case, the relay is a legitimate user or acts like a legitimate user who will help to counter external eavesdroppers and increase the security of the networks. In the literature, one-way trusted relaying (e.g., [2]–[6]) and two-way trusted relaying (e.g., [7]–[10]) are both studied. In the case of untrusted relay, the relay node acts both as an eavesdropper and a helper, i.e., the eavesdropper is co-located with the relay node. For this case, it is important to find out whether the untrusted relay is still beneficial. One-way untrusted relaying is considered in [11]–[14]. A This work is supported by the Program for New Century Excellent Talents in University (NCET) under grant NCET-11-0331 and National 973 project under grant 2012CB316100.

978-1-4673-5939-9/13/$31.00 ©2013 IEEE

destination-based jamming (DBJ) technique was proposed in [11] without source-destination link. The performance of DBJ in fading channel and multi-relay scenarios was analyzed in [12]. When the source-destination link exists, authors in [13] discussed whether cooperating with the untrusted relay is better than treating it as a passive eavesdropper. In [14], joint source and relay beamforming design was proposed for MIMO one-way untrusted relay systems. Authors in [15] considered the two-way untrusted relay system and a Stackelberg game between the two sources and the external friendly jammers was formulated as a power control scheme. In view of all these existing literature, the problem of secure beamforming for MIMO two-way untrusted relaying has not been considered yet. In this paper, we investigate a MIMO two-way relay system, where the two sources exchange information with each other through an untrusted relay. The problem is more interesting than the case of one-way relay systems, since by applying network coding, the relay only needs to decode the network-coded message rather than each individual message and hence the network coding procedure itself also brings certain security. If incorporating the multipleinput multiple-output (MIMO) techniques into two-way relaying, the unique and inherent nature of MIMO two-way relaying for physical layer security is that we can align the received signals at the relay via transceiver design, since the aligned signals do not affect the desired destinations (by selfinterference cancelation) but can be against eavesdropping. Such an idea is similar to the so-called signal alignment technique [16], which also motivates the research of this paper. In [16], the signal alignment technique was used to increase the throughput of the MIMO Y channel while in this paper, it is used to enhance the security of the MIMO two-way untrusted relay system. In this paper, we study the joint source and relay beamforming designs in MIMO two-way untrusted relay systems for enhancing physical layer security. We propose an iterative method to decouple the problem into several subproblems that can be solved in an alternating manner. Moreover, we conduct asymptotical analysis when the powers of the source or/and relay nodes approach zero or infinity. Specifically, when the powers of the source and relay nodes approach infinity, the maximum secrecy rate can be achieved if the transceiver beamformers are designed such that the received signals at the relay can be aligned to be parallel.

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at relay is,



GA 

R

HB

HA TA

A

r yR = HA qA sA + HB qB sB + nR .

GB

r In the second phase, the relay amplifies yR by multiplying it NR ×NR with a precoding matrix F ∈ C . The transmit signal vector from the relay node is expressed as



B

r xrR = FyR .

TB

Fig. 1.

(1)

(2)

Then the received signal at the source node i during the second phase is,

MIMO two-way relay model.

Notations : Tr(A) and A−1 denote the trace and inverse of matrix A, respectively. ∥q∥ denotes the norm of the vector q. σmax (A) is the largest singular value of A. λmax (A) and ψ max (A) are the largest eigenvalue and the corresponding eigenvector of A, respectively. λmax (A, B) and ψ max (A, B) are the largest generalized eigenvalue and the corresponding generalized eigenvector of the matrices A and B, respectively. AN = I − AH (AAH )−1 A is the projection matrix onto the null space of A. [x]+ denotes max (0, x). Pi (i ∈ {A, B, R}) represents the transmit power of node i. Throughout this paper, n denotes the zero mean circularly symmetric complex Gaussian noise vector with n ∼ CN (0, I). II. S YSTEM M ODEL We consider a two-way relay system as shown in Fig. 1, where A and B are the source nodes that exchange information with each other with the assistance of the relay node R. The relay acts as both an essential helper and a potential eavesdropper, but does not make any malicious attack. Therefore, the relay cooperates with the sources to relay their messages but try to decode the messages in the process of relaying. It is assumed that the relay adopts the amplify-and-forward (AF) strategy due to its low implementation complexity and mathematical tractability. It is worth mentioning that the full decode-and-forward (DF) relay strategy is not applicable for the untrusted relay, since its needs the signals received from the sources to be fully decoded at the relay which is against secure transmission. We also assume that full channel state information (CSI) is available at all nodes. This assumption is commonly used in the literature [12]–[15]. The number of antennas of A, B and R are denoted as NA , NB and NR , respectively. As shown in Fig. 1, TA ∈ CNB ×NA , TB ∈ CNA ×NB , HA ∈ CNR ×NA , GA ∈ CNA ×NR , HB ∈ CNR ×NB , GB ∈ CNB ×NR denote the channel matrices of link A → B, B → A, A → R, R → A, B → R and R → B, respectively. In this paper, we only consider rank-one precoding at the sources. Let qA ∈ CNA ×1 and qB ∈ CNB ×1 be the beamforming vectors of A and B, respectively. The transmitted symbol at the source node i is denoted by si ∈ C with E(|si |2 ) = 1, i ∈ {A, B}. In the two-way relay scheme, two phases are needed to complete one round of information exchange between the two source nodes. In the first phase, A and B simultaneously transmit signals to the relay node R and the received signal

˜ ir = Gi xR + ni y

(3)

= Gi FHi qi si + Gi FH¯i q¯i s¯i + Gi FnR + ni , i ∈ {A, B}, where ¯i = {A, B} \ i. Subtracting the back propagated selfinterference term Gi FHi qi , each source node obtains the equivalent received signals: yir = Gi FH¯i q¯i s¯i + Gi FnR + ni , i ∈ {A, B}. The information rate from node i to node ¯i is ( ) 1 H H H −1 Rri¯i = log2 1 + qH i Hi F G¯i K¯i G¯i FHi qi . 2 where K¯i = G¯i FFH G¯H i + I.

(4)

(5) (6)

For the untrusted relay to eavesdrop the messages from both source nodes, it tries to fully decode the two signals sA and sB . Therefore, the information sum-rate achieved at the untrusted relay can be expressed as the maximum sum-rate of a two-user MIMO multiple-access channel, given by r RrR , I (yR ; sA , sB ) 1 H H H = log2 I + HA qA qH A HA + HB qB qB HB . (7) 2 After simple matrix manipulation, we can obtain the achievable secrecy sum rate [17] given on the top of the next page. Our goal is to maximize the secrecy sum rate by jointly optimizing the beamformers F, qA and qB subject to the source and relay power constraints. An optimization problem can be formulated as

Rrmax = max Rrs

F,qA ,qB subject to ∥qi ∥2 ≤ Pi , ( H H Tr FHA qA qH A HA F

(9) i ∈ {A, B}, ) H H H ≤ PR . + FHB qB qH B HB F + FF

Here, Rrmax denotes the maximum secrecy sum rate of relaying. For the purpose of comparison, the two-way direct transmission scheme is also considered, where the source nodes A and B transmit directly to the other in the first and second phases, respectively, and the relay node is treated as a pure eavesdropper. The maximal achievable secrecy sum rate of the direct transmission, denoted as Rdmax is given by ∑ 1[ ( )]+ H . Rdmax = log2 λmax I + Pi TH i Ti , I + Pi Hi Hi 2 i∈{A,B}

(10) For brevity, the detailed derivation of direct transmission scheme is omitted here and can be found in [14], [18].

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+

Rrs = [RrAB + RrBA − RrR ] (8) [ ]+ ( )( ) H H H H −1 H H H H −1 1 + qA HA F GB KB GB FHA qA 1 + qB HB F GA KA GA FHB qB 1 . = log2 H H H H H H H H H H H 2 1 + qA HA HA qA + qH B HB HB qB + qA HA HA qA qB HB HB qB − qB HB HA qA qA HA HB qB

III. S ECURE B EAMFORMING D ESIGN

A. Low Relay Power

First, we present the optimal structure of the relay beamforming matrix F for given source beamforming vectors. Define the following two QR decompositions: H r [GH A GB ] = VR1 ,

(11)

[HA qA HB qB ] = URr2 .

(12)

where V ∈ CNR ×min{NA +NB ,NR } , U ∈ CNR ×2 are orthonormal matrices and Rr1 , Rr2 are upper triangle matrices. Then, we can obtain the optimal structure of the relay beamforming matrix F to simplify the problem (9) based on the following lemma. Lemma 1. The optimal relay beamforming matrix F that maximizes the secrecy sum rate has the following structure: F = VAUH ,

(13)

where A ∈ Cmin{NA +NB ,NR }×2 is an unknown matrix. Proof: Note that the relay beamforming matrix F only influences RrAB and RrBA . Therefore, the optimal F is actually the same as the F that maximizes RrAB + RrBA . Due to the rank-one precoding at each source node, we have the equivalent channel Hi qi from source node i to relay. Therefore, using the results in [19, Theorem 1], we readily have Lemma 1. Note that the number of unknowns in F is reduced from NR2 to 2 min{NR , NA + NB } after applying Lemma 1. It is not easy to find the optimal solution to the problem (9). Even after substituting the optimal structure of F (13) into (8), the problem is still nonconvex since the secrecy sum rate Rrs is not a convex function of qA , qB and A. Therefore, we optimize the source beamforming vectors qA , qB and the unknown matrix F in an alternating manner. Specifically, given qA and qB , we use the gradient method to search A [19]. Then, given F and qi , we can find the optimal q¯i [20, Appendix B]. This iterative process is repeated until converged. The details of the algorithm is omitted here and can be found in [20, Section III]. IV. A SYMPTOTIC A NALYSIS The optimization problem (9) is nonconvex and thus difficult to obtain the globally optimal solution. However, as the powers approach zero or infinity, we can get some asymptotic results. These results reveal some characteristics of this problem and give important insights into designing beamforming vectors qA and qB .

In this subsection, we present the asymptotic maximum secrecy sum rate when relay power approaches zero. As the relay power approaches zero, the information rate through the relay link approaches zero, which means that RrAB + RrBA approaches zero. On the other hand, the information rate leaked to the untrusted relay RrR is not related to the relay power and thus does not approach zero. Hence, the maximum secrecy sum rate of the two-way relay scheme is zero when PR → 0. Proposition 1. When the relay power PR → 0, Rdmax ≥ Rrmax ,

(14)

Proof: Since Rdmax is not related to the relay power PR and Rrmax → 0 as PR → 0. we obtain this proposition. B. High Relay Power In this subsection, we let relay power approach infinity and present the cases of both high and low source powers. 1) Low source powers case: PA → 0 and PB → 0: Lemma 2. When PA → 0 and PB → 0, the maximum secrecy sum rate of the two-way direct transmission scheme is, ∑ [ ( )]+ 1 H Pi λmax TH Rdmax ≈ i Ti − Hi Hi 2 ln 2 i∈{A,B}

Proof: It is easily obtained from [18, Corollary 2]. Lemma 3. When PR → ∞, PA → 0 and PB → 0, the optimal source beamforming vectors of the two-way relay scheme are ( ) √ H Piψ max HH i H¯i H¯i Hi ( ) , qi = (15) H ψ max HH ∥ψ i H¯i H¯i Hi ∥ and the maximum secrecy sum rate is Rrmax ≈

( ) 1 H PA PB λmax HH A HB HB HA . 2 ln 2

(16)

Proof: Please see [20, Appendix E]. Proposition 2. When PR → ∞, PA → 0 and PB → 0, we have Rdmax ≥ Rrmax .

(17)

Proof: This proposition can be easily obtained from Lemma 2 and Lemma 3. As PA → 0 and PB → 0, Rrmax approaches zero faster than Rdmax . Thus, the proof of Proposition 2 is completed.

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2) High Source Powers case: PA → ∞ and PB → ∞: Lemma 4. When PA → ∞ and PB → ∞, the maximum secrecy sum rate of the two-way direct transmission scheme is, ∑ Rdmax ≈ Ωi ,

Lemma 6. When PR → ∞, PA → ∞, PB → ∞, and NA ≤ NR , NB ≤ NR , NA + NB > NR , the maximum secrecy sum rate of the two-way relay scheme approaches infinity while the maximum secrecy sum rates of two-way direct transmission approaches a constant. Thus, Rrmax ≥ Rdmax .

i∈{A,B}

(19)

Proof: It can be obtained from Lemma 4 and Lemma 5. where  [ ( )] 1 + H   log2 λmax TH if Ni ≤ NR i Ti , Hi Hi 2 Ωi = .Remark 2. As shown in Lemma 4 for two-way direct transmis[ ( )]   1 log Pi + log λmax HN TH Ti HN sion, in the infinite power case, the infinite maximum secrecy if N > N i R 2 2 i i i 2 sum rate needs NA > NR or NB > NR . Lemma 6 reveals that with the signal alignment technique at the untrusted relay, Proof: This lemma is based on [14, Lemma 7]. Here, we the infinite maximum secrecy sum rate only needs NA +NB > assume that the entries of the channel matrices are generated NR , which lowers the requirement of the numbers of antennas m×n from continuous distribution. As a result, Rank(H ) ≥ at the two source nodes. The result clearly demonstrates the N H min{m, n} with probability one and the condition Hi Ti ̸= benefits of signal alignment for physical layer security, which 0 is also satisfied with probability one. is the unique feature in MIMO two-way relaying. Lemma 5. When PR → ∞, PA → ∞ and PB → ∞, the maximum secrecy sum rate of the two-way relay scheme is, V. S IMULATION Rrmax ≈  1 1   log2  2,  2  1 − (σmax (UH A UB ))

In this section, we perform simulation for all the cases discussed in Section IV. In the simulation, we assume that the if NA + NB ≤ NR channel reciprocity holds, i.e., HA = GTA , HB = GTB and TA = TTB . The following example of channel coefficients ∥HA qA ∥2 ∥HB qB ∥2 1   log , otherwise max  2  realization (every entry of the matrices is generated from (qA ,qB )∈ 2  ∥HA qA ∥2 + ∥HB qB ∥2 {β∈R, βHA qA =HB qB } CN (0, 1) distribution) are used to show the asymptotical NR ×min{NA ,NR } NR ×min{NB ,NR } performance. where UA ∈ C and UB ∈ C are obtained from the QR decomposition of HA and HB ,   respectively, i.e., 0.27 − 0.10i 0.13 − 1.24i 0.60 + 0.83i  0.95 + 0.87i −0.45 + 0.22i −0.46 + 0.35i  Hi = Ui Ri , i ∈ {A, B}, (18)   ¯ A =  0.41 − 0.76i −0.67 − 0.74i −0.01 + 1.06i (20) H    −1.00 + 0.26i −1.59 − 0.95i −0.45 − 0.30i  where Ri ∈ Cmin{NR ,Ni }×Ni are upper triangle matrices. −1.14 + 0.11i −0.52 − 0.06i 0.16 + 0.01i Proof: Please see [20, Appendix G]. Remark 1. It is worth mentioning that in Lemma 5, the optimal beamforming design is similar as the so-called signal alignment [19]. Specifically, Ui is the orthonormal basis of the column space of Hi . If NA + NB > NR , there is intersection subspace between the column spaces of Hi with probability one and thus there exists β ∈ R such that βHA qA = HB qB . The maximum secrecy sum rate goes to infinity as the source powers approach infinity. On the other hand, if NA +NB ≤ NR , there is no intersection subspace with probability one and there is an upper( bound maximum ( for the )) secrecy sum rate. Actually, arccos σmax UH U , which B A is the minimum angle between any two vectors from the the respect two column spaces, can be viewed as the angle between these two column spaces. From Lemma 4 and Lemma 5, we find that it is difficult to compare these schemes when PR → ∞, PA → ∞ and PB → ∞. Actually, the results of the comparison depend not only on NA , NB , NR but also on the channel conditions. In the following, we only present the comparison result in one interesting case.



0.36 + 0.71i  0.62 − 0.35i  ¯ B =  −0.48 − 0.35i H   −0.29 + 1.53i −0.07 + 0.14i

−0.05 − 1.12i 0.22 + 0.87i 0.28 + 0.30i −0.26 + 0.87i 0.15 + 0.93i

 0.62 − 1.66i −0.85 − 0.18i   −0.37 + 1.69i  (21) −1.68 + 0.02i  0.90 − 0.39i



 0.05 + 1.36i 1.11 − 0.57i −0.52 − 0.07i ¯ A =  0.92 − 0.94i −0.57 − 1.17i −0.40 − 0.64i (22) T −0.06 − 1.30i −0.93 + 0.12i 0.15 + 0.40i If the channel matrix we need is smaller than the above matrices, we simply choose the left upper part of the corresponding matrix. For instance, if NA = 2, NR = 3, we choose ¯ A (1 : 3, 1 : 2). In all figures, DT and TWR represent HA = H two-way direct transmission and relay scheme, respectively. Note that the iterative algorithm in Section III is not guaranteed to find the optimal solution and the convergent point may be far from the optimal solution. In the simulation, we choose the initial point as the asymptotic optimal beamforming vectors qA and qB obtained from the asymptotic analysis.

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3.5

2

DT Upper bound of DT TWR TWR (Signal alignment)

DT 1.6

Maximum secrecy sum rate (bps/Hz)

Maximum secrecy sum rate (bps/Hz)

1.8

TWR

1.4 1.2 1 0.8 0.6 0.4

3

2.5

2

1.5

1

0.5

0.2 0 −20

0 −15

−10

−5

0

5

10

15

20

0

2

4

6

Fig. 2. Comparison of the two schemes with low relay power. NA = 2, NR = 3, NB = 2 and PR = −20 dB.

12

14

16

18

20

9

1

DT DT, asymptotic TWR TWR, asymptotic

Maximum secrecy sum rate (bps/Hz)

8

Maximum secrecy sum rate (bps/Hz)

10

Fig. 4. Comparison of the two schemes in high power regimes when NA = 2, NR = 3, NB = 2 and PR = 40 dB.

10

0

10

−1

10

−2

10

DT DT, asymptotic 2P 2P, asymptotic

−3

10

7

6

5

4

3

2

1

0

−4

10 −20

8

PA = PB (dB)

PA = PB (dB)

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

0

2

4

6

8

10

12

14

16

18

20

PA = PB (dB)

PA = PB (dB)

Fig. 3. Comparison of the two schemes with high relay power when NA = 2, NR = 3, NB = 2 and PR = 40 dB.

Fig. 5. Comparison of the two schemes in high power regimes when NA = 3, NR = 2, NB = 3 and PR = 40 dB.

A. Low Relay Power

C. High Relay Power and High Source Powers

In Fig. 2, we compare the two schemes when the relay power is as low as -20dB. We find that the maximum secrecy sum rate of the relay scheme is zero and direct transmission is better than the relay scheme, which verifies Proposition 1.

Case 1) NA = 2, NR = 3, NB = 2: Fig. 4 clearly shows the importance of signal alignment for security. We see that in Fig. 4 the maximum secrecy sum rate of relay scheme goes to infinity when the source powers tend to infinity, while that of the direct transmission scheme is upper bounded. In this channel setup, the upper bound of the maximum secrecy sum rate of the direct transmission scheme is about 1.82bps/Hz according to Lemma 4. Case 2) NA = 3, NR = 2, NB = 3: In this case, Ni > NR . In Fig. 5, both maximum secrecy sum rates of the two schemes approach infinity as the powers tend to infinity. We also find that the asymptotical results are quite accurate. In this case, although the signal alignment can be achieved, the

B. High Relay Power and Low Source Powers Fig. 3 shows the performance of the two schemes when PR = 40dB and the source powers are low. We find that the relay scheme is much worse than the direct transmission r scheme. By careful observation, we see that Rmax decreases d to zero as twice faster as Rmax when the source powers tend to zero. Thus, the Proposition 2 is validated.

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R EFERENCES

1.4

Maximum secrecy sum rate (bps/Hz)

1.3

DT Upper bound of DT TWR, P =40dB

1.2

R

1.1

TWR, PR=50dB Upper bound of TWR

1

0.9

0.8

0.7

0

5

10

15

20

25

30

PA = PB (dB)

Fig. 6. Comparison of the two schemes in high power regimes when NA = 2, NR = 5, NB = 2 and PR = 40 dB.

direct transmission scheme is better than the relay scheme. Actually, when NA > NR and NB > NR , the comparison result depends on the channel realization and we can not conclude which scheme is better. Case 3) NA = 2, NR = 5, NB = 2: Finally, we show the scenario when NA +NB ≤ NR in Fig. 6. Under this condition, both schemes have upper bounds for their maximum secrecy sum rates. We find that the two-way relay scheme is better in this channel setup. We also plot the relay scheme when PR = 50dB. The curve can approach the upper bound more closely than the curve when PR = 40dB, which implies that to approach the upper bound given in Lemma 5, we need the powers of all the three nodes go to infinity and the relay power should be much larger than the source powers. In summary, we conclude that all the derived asymptotical results are very accurate and in accord with the numerical results. VI. C ONCLUSION In this paper, we investigated a MIMO two-way AF relay system where the two source nodes exchange information with an untrusted relay. For two-way relay schemes, we proposed an efficient algorithm to jointly design the source and relay beamformers iteratively. Furthermore, we analyzed the asymptotical performance of the secure beamforming schemes in low and high power regimes of the sources and relay. Simulation results validated our asymptotical analysis. From these results, we conclude that the conventional twoway direct transmission is preferred when the relay power goes to zero. When the relay power approaches infinity and source powers approach zero, the direct transmission scheme also performs better. Moreover, when NA ≤ NR , NB ≤ NR , NA +NB > NR and the relay and source powers go to infinity the two-way relay scheme achieves the best performance by using the signal alignment technique.

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